The polynomial of degree 5, P(x) has leading coefficient 1, has roots of multiplicity 2...

Since each root is a linear factor, we can write:
${\displaystyle P\left(x\right)=x^2{(x-1)}^2(x+3)}$ 
${\displaystyle =x^2(x^2-2x+1)(x+3)}$ 
${\displaystyle =x^5+x^4-5x^3+3x^2}$ 
Any polynomial that contains these zeros and at least these multiplicities is going to be a multiple (scalar or polynomial) of this P(x) 
Answer: 
${\displaystyle P\left(x\right)=x^5+x^4-5x^3+3x^2}$